Competition · AMC preparation · step 4 of 4
AMC 8 · 2011 · #15
Grade 6 arithmeticPick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Computing 4⁵ · 5¹⁰ as a brute number is painful, but the bases 4 and 5 are hiding a 2 inside 4. Tool #16 (Change Focus) says: rewrite 4 as 2² so the bases become 2 and 5. Now the 2s and 5s pair up perfectly into 10s, turning the product into a clean power of 10. Tool #9 (Easier Related Problem) then takes over: counting digits in 10¹⁰ is far easier than in 4⁵ · 5¹⁰, and we already know the rule 10ⁿ has n+1 digits.
Rewrite 4 as a power of 2
Since 4 = 2², the power-of-a-power rule gives 4⁵ = (2²)⁵ = 2¹⁰.
Changing the base from 4 to 2 lets the exponents talk to the 5¹⁰ next door.
6.EE.A.1Change Focus Count The ComplementCombine into a power of 10
Now both factors share exponent 10, so 2¹⁰ · 5¹⁰ = (2·5)¹⁰ = 10¹⁰.
Every 2 finds a partner 5 and together they form a 10 — the whole product collapses into 10¹⁰.
The product 4⁵ · 5¹⁰ can be rewritten as the single power 10¹⁰.
▸ Why?
First 4⁵ turns into 2¹⁰, so the product becomes 2¹⁰ · 5¹⁰, two powers that now share the exponent 10.
▸ Why?
Since 4 = 2 · 2, writing out 4⁵ is five copies of 2 · 2, which is ten 2's multiplied together, namely 2¹⁰.
▸ Why?
Counting those 2's means taking five groups of two, and five groups of two is ten.
▸ Why?
The ten 2's can be gathered into one product in any grouping without changing the result.
▸ Why?
Now 2¹⁰ is ten 2's and 5¹⁰ is ten 5's, so each 2 can be set beside a 5 to make ten copies of 2 · 5 = 10, that is 10¹⁰.
▸ Why?
The factors can be reordered so that every 2 stands next to a 5.
▸ Why?
Each neighbouring 2 and 5 can be regrouped into a single factor (2 · 5) ten times without changing the product.
Find the digit pattern
Switch to an easier question — the digits of 10¹⁰; the rule is 10ⁿ has n+1 digits, a 1 followed by n zeros.
Multiplying by 10 tacks on a zero — a Grade 5 place-value pattern.
5.NBT.A.2Solve An Easier Related ProblemApply the digit rule
Apply n = 10: 10¹⁰ has 10 + 1 = 11 digits.
One leading 1 plus ten trailing zeros equals eleven digits total.
5.NBT.A.2Solve An Easier Related ProblemWhenever you see 2s and 5s with matching exponents, pair them into 10s — the answer just falls out as a power of 10.
- Rewrite 4 as a power of 2
- Combine into a power of 10
- Find the digit pattern
- Apply the digit rule
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